[SOLVED] Intermediate Microeconomics UA10 Homework 10 Fall 2025

Intermediate Microeconomics UA10

Homework 10 Fall 2025

NOT GRADED: Will be solved in the final review session

1. This question is entirely geometric. Work throughout with the tracking problem:

u(a, ω1) = 1, u(a, ω2) = 0, u(b, ω1) = 0, u(b, ω2) = 1,

and prior

µ1 = Pr(ω1) = 0.5.

Use graph paper. Unless otherwise indicated, place p1 ∈ [0, 1] on the horizontal axis and utilities/costs on the vertical. Your figures should follow the lecture diagrams. No algebra is required beyond labeling.

(a) Draw the expected utilities of a and b and the upper envelope

Uˆ(p1) = max{p1, 1 − p1}.

Use solid for the envelope and dashed for dominated segments.

(b) On a new set of axes, draw a strictly convex, symmetric cost curve C(p1) with

C(0.5) = 0,

and which becomes very steep as p1 → 0 or p1 → 1. (Use any smooth “Shannon-like” shape.)

(c) Consider an experiment with posteriors

γL1 = 0.40, γ1
H = 0.90.

Solve for the Bayes weight P
L
, then illustrate the expected cost:

Cost = P
L C(γ1
L
) + (1 − P
L
) C(γ1
H).

(d) In a third figure, construct geometrically the net-utility functions

Na(p1) = Ua(p1) − C(p1), Nb(p1) = Ub(p1) − C(p1),

and their net-utility envelope

Nˆ(p1) = max{Na(p1), Nb(p1)}.

As in class, use a dashed line for the dominated portions of the net utility func-tions.

(e) On the same figure, plot

(γ1
L
, Nˆ(γ1
L
)), (γ1
H, Nˆ(γ1
H)),

and draw the chord joining them. Label the height at the prior p1 = 0.5.

(f) Explain briefly, using your picture, why the experiment

{γ1
L
, γ1
H} = {0.40, 0.90}

cannot be optimal.

(g) Illustrate the optimal experiment on the same axes and explain its geometric features.

2. Now return to the symmetric tracking problem but introduce stakes x =
1/3:

u(a, ω1) = u(b, ω2) = 1
3
, u(a, ω2) = u(b, ω1) = 0,

with prior µ1 = 0.5. Assume learning incurs quadratic cost:

C(p1) = (p1 − 0.5)2
.

(a) Compute the optimal distance d
ˆ that maximizes expected net utility and the optimal posteriors

γL1 = 0.5 − ˆd, γ1
H = 0.5 + ˆd.

(b) Compute separately the value of learning (the gain in Uˆ) and the cost of learning at d
ˆ.

(c) Draw (on one diagram):

• net-utility curves Na(p1) and Nb(p1),

• their envelope Nˆ(p1),

• the two optimal posteriors,

• the chord between them,

• and the vertical gap at the prior p1 = 0.5 representing maximized net utility.